94 Appendix to a Method of extending 
And, fmce zz is = and ee is = it follows 
1 36 x y 3 6' 
that - will be = Therefore - will be = 4, and 
j Z 10 / Z 6 £ 4 I I \ I J / Z 10 Z 4 1 jv 
and -73- (— ~6 x ~i— yy x ?) ~~ T4T 5 7^ ( —77 x 7— 777 x 7) — 
Therefore the feries — + ?l! + g£ C . -will 
7 x 8 7 * 
, -1 • r i, ^ C , G t JL« P t0 .111. 111, 111, BcCt 
in. this cafe be = -+— + — + — r- + &c. — — - — - — - — + 
3 27 243 2187 3 
.023 , 472,031, &c. + .011,690,017, &c. + .007,414, 54 2 > &c. + g cc _ 
27 243 2187 
.0 37, 037, 037, 8ce. + . ooo, 869, 334, 8cc. + 000,048,1 07, 
&c. + 000,003,390, &x. + &c. =.037,957,868, 8cc. 
Therefore jp/ 3 e x the feries ~+^ 7 -+^ 3 .+? 7 ! + See. 
is equal to — x .037,957,868, &c. = : - 5I ’\ 31 ’ 472 ’ 
^ 1,817,121 041^011 J I.8l7,I2I 
-083,556,06. 
Confequently the whole expreffion \/ 3 (77 + s/tfe-z + 
4v/ 3 ^ x the feries — + ^- + ^ + ^ + 8cc. is = 
* eeeee 
1. 053,601,31+. 083, 556, 06 = 1. 137, 157, 37 ; that is, 
the root of the propofed equation x 3 - x — } is = 
1 * I 37> I 57>37* Q. E. I. 
Art. 12. This value of the root of the equation 
x 3 -x = \ is exatft to fix places of figures, the more ac- 
curate value of it being 1.137,158,164. We may 
therefore conclude that the expreffion here made ufe of 
to determine the value of x t to wit, ■s/ 3 \e+'z + \/ 3 [ e-z+ 
